Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups

TL;DR

This paper derives explicit formulas for geodesics in sub-Lorentzian and Finsler problems on low-dimensional Lie groups using new convex trigonometric functions, advancing the understanding of extremals in these geometric contexts.

Contribution

It introduces new convex trigonometric functions and provides explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups.

Findings

Explicit formulas for extremals in sub-Lorentzian and Finsler problems.

Introduction of convex trigonometric functions $ ext{cosh}_ abla$ and $ ext{sinh}_ abla$.

Generalization of classical hyperbolic functions to unbounded convex sets.

Abstract

In this paper, we consider the problem of finding geodesics in a series of left-invariant problems endowed with sub-Lorentzian and Finsler structures. Explicit formulas for extremals are obtained in terms of convex trigonometric functions. In the sub-Lorentzian setting, the new trigonometric functions $\cosh_\Omega$ and $\sinh_\Omega$, developed here, prove especially useful; they generalize the classical $\cosh$ and $\sinh$ to the case of an unbounded convex set $\Omega\subset\mathbb{R}^2$.